Geometric price indices as a best predictor

A geometric index can be motivated in the same way as an arithmetic index. Formally, a geometric index is the value \(I^{G}\) that solves \[\begin{align*} \min_{I} E\left[\left(\log\left(\frac{p_{t}}{p_{0}}\right) - \log(I) \right)^{2}\right], \end{align*}\] the solution to which is \(I^{G} = \exp(E[\log(p_{t} / p_{0})])\). But this is just a geometric index, as \[\begin{align*} \exp\left(E\left[\log\left(\frac{p_{t}}{p_{0}}\right)\right]\right) &= \exp\left(\sum_{i = 1}^{n} P_{i} \log\left(\frac{p_{it}}{p_{i0}}\right)\right) \\ &= \prod_{i = 1}^{n} \left(\frac{p_{it}}{p_{i0}}\right)^{P_{i}}, \end{align*}\] with weights equal to the probability of observing a price relative. It is worth noting that, due to the logarithms in the prediction problem, a geometric index can equally be motivated as finding a value \(I^{G}\) such that \(I^{G} \times p_{i0}\) best predicts \(p_{it}\) (or \(p_{it} / I^{G}\) best predicts \(p_{i0}\)). This is a property not shared with arithmetic indices. Whereas an arithmetic index predicts price changes over time, a geometric index takes the price for a good in period 0, inflates/deflates it with the price index, and uses the result to predict the price for that good in period \(t\). Consequently, a geometric index fits more neatly into the stochastic approach as it directly corresponds with how a price index is used to inflate and deflate prices over time in practice.25

The Törnqvist index usually comes out as the best index in the stochastic approach because it picks a sensible value to represent the probability of observing a price relative. The idea behind the Törnqvist probabilities is that the likelihood of observing a particular price relative for a good depends on the expenditure/revenue share of that good across both periods. Thus, the Törnqvist index sets the probabilities as average expenditure shares: \[\begin{align*} P_{i} = \frac{1}{2} \frac{p_{i0}q_{i0}}{\sum_{j = 1}^{n} p_{j0}q_{j0}} + \frac{1}{2} \frac{p_{it}q_{it}}{\sum_{j = 1}^{n} p_{jt}q_{jt}}. \end{align*}\] The disadvantage of these probabilities is that they require information for both period-0 and period-\(t\) expenditure/revenue shares, something that may not be known at the time the index is calculated.

  1. Motivating a price index as solving the problem \(\min_{I} E[(p_{t} - p_{0} I)^{2}]\) results in \(I = E(p_{t} p_{0}) / E(p_{0}^{2})\). If \(P_{i}\) is proportional to \(q_{it} / p_{i0}\) then this reduces to a fixed-basket index, emphasizing that the representation of the prediction problem for an index number also depends on what determines the probability of observing a transaction.↩︎